How Can I Merge a Separate Edge Loop into a Face?
Knife Project

If the vertices and box are two separate objects, great! If not, select one vertex of the edge loop, key Ctrl + L to Select Linked, key P to open the Separate menu, and select By Selection.

Move the loop away from the object on the Y-Axis a bit. Key Numpad 5 to enter Orthographic View. Key most likely either Numpad 3 or Ctrl + Numpad 3 (whatever view shows the loop in front of the box).

Select the edge loop, then the box, then key Tab to enter Edit Mode for the box. Select the one face you wanted to add the vertices to, key Ctrl + I to Select Inverse, and key H to hide the other faces. Key Space, type Knife Project, and key Enter. Key Alt + H to reveal the other faces.